Circles of Apollonius two ways

TL;DR

This paper develops two methods using enriched enumerative geometry to count Apollonius circles consistently over various fields, ensuring invariance of the number of solutions beyond the real numbers.

Contribution

It introduces two novel approaches to count Apollonius circles with invariance over any field of characteristic not 2, expanding the classical real-number results.

Findings

Two methods for counting Apollonius circles with invariant solutions

Extension of Apollonius problem to arbitrary fields of characteristic not 2

Discussion of geometricity problem for local indices in enriched enumerative geometry

Abstract

Because the problem of Apollonius is generally considered over the reals, it suffers from variance of number: there are at most eight circles simultaneously tangent to a given trio of circles, but some configurations have fewer than eight tangent circles. This issue arises over other non-closed fields as well. Using the tools of enriched enumerative geometry, we give two different ways to count the circles of Apollonius such that invariance of number holds over any field of characteristic not 2. We also pose the geometricity problem for local indices in enriched enumerative geometry.